Can a quaternion represent more than 360 degrees of rotation?

I like Euler rotation based on degrees so I can keyframe an animation of a large number of degrees, like 1000, and get multiple full 360-degree revolutions on the object for purposes of setting it spinning, etc.

However I'm at a point now when I was working on something that involved a lot of consecutive animations among two different axes that I think I'm seeing the effects of gimbal lock. When I perform one of the chained animations, my object keeps flipping 180 degrees after it completes a 90-degree rotation. Does this sound like gimbal lock? It only happens after other prior and successful rotations along different axes.

In any case, I tried switching to Quaternions but it appears that quaternions aren't suited for this spinning effect, since they just get you an orientation and they treat a multiple of 360 as 0 using this:

I like Euler rotation based on degrees so I can keyframe an animation of a large number of degrees, like 1000

Why do you need keyframes far enough apart that this is necessary? Generally speaking, keyframes should only be at most 1/10th of a frame apart. You shouldn't be able to spin 1080 degrees in less than one tenth of a second.

Sorry, you misunderstand. I can set a single animation to last, say, 10 seconds, and in that 10 seconds I get 5 full revolutions, or 360*5 degrees. That is what I meant. But I gather this is not possible with quaternions.

You're simply doing it the wrong way. If you have a single animation that lasts 10 seconds and during that time, you have a 5 360 degree revolutions, then what you have is a single animation that lasts 10 seconds, where every 1/30th of a second, you rotate the object 6 degrees.

... What are you talking about? I was the one who suggested that you rotate it 6 degrees every 1/30th of a second. I didn't say that it was wrong.

What's wrong is thinking of the rotation as 1800 degree in 10 seconds. They're both functionally the same, but the smaller one uses smaller numbers, so you don't have to deal with rotations that quaternions can't handle.

The way you're doing it is to attempt to create a quaternion for a 1800 degree rotation, then dividing that quaternion by time. You instead should be dividing the rotation by the time, and constructing a quaternion to rotate the object based on how much time elapsed since the last time you did that. You update the orientation by the delta quaternion.